Subgroups — #15778295

Order 3328.

296 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718055	C₂	1	1	yes	maximal, characteristic
2	#13116441	C₂	1	2	yes	maximal
3	C₄×#88052	C₂	1	1	yes	maximal, characteristic
4	#13112922	C₂	1	2	yes	maximal
5	C₁₃×#1772	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₃×#108	C₄	1	1	yes	characteristic
7	C₁₃×#427	C₄	1	1	yes	characteristic
8	C₁₃×#405	C₂²	1	2	yes	—
9	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
10	C₂×#88052	C₂²	1	2	yes	—
11	C₂×#88052	C₂²	1	1	yes	characteristic
12	C₂×#88052	C₂²	1	1	yes	characteristic
13	C₁₃×#339	C₄	1	2	yes	—
14	#11718058	—	2	4	no	—
15	#11718055	C₄	1	2	yes	complemented
16	#11718055	C₄	1	2	yes	complemented
17	#11718053	C₄	1	2	yes	complemented
18	C₄×#6090	C₄	1	2	yes	complemented
19	C₂×C₁₃×M₁₆	C₂×C₄	1	2	yes	—
20	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
22	C₁₃×#117	C₂×C₄	1	2	yes	—
23	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
24	C₁₃×#108	—	2	1	no	—
25	C₁₃×#108	—	2	1	no	—
26	C₁₃×#107	—	2	1	no	—
27	C₁₃×#106	—	2	1	no	—
28	C₁₃×#97	—	2	2	no	—
29	C₄×C₈×C₁₃	—	2	2	no	—
30	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
32	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
33	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
34	#88052	—	2	2	no	—
35	#88052	—	2	1	no	—
36	#88052	—	2	1	no	—
37	#88052	C₂×C₄	1	4	yes	—
38	#88052	D₄	1	2	yes	—
39	#88052	Q₈	1	2	yes	—
40	C₂×#6090	—	2	2	no	—
41	C₂×#6090	C₂×C₄	1	2	yes	—
42	C₂×#6090	C₂×C₄	1	2	yes	—
43	#12667	—	13	1	no	maximal, Sylow
44	C₂²×C₄×C₁₃	—	2	1	no	—
45	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
46	C₁₃×M₁₆	—	2	2	no	—
47	C₁₃×M₁₆	—	2	2	no	—
48	C₂×C₈×C₁₃	—	2	2	no	—
49	C₂×C₈×C₁₃	C₄²	1	1	yes	characteristic
50	C₂×C₈×C₁₃	C₄²	1	1	yes	characteristic
51	C₂×C₈×C₁₃	M₁₆	1	1	yes	characteristic
52	C₂×C₈×C₁₃	M₁₆	1	1	yes	characteristic
53	C₂×C₈×C₁₃	M₁₆	1	1	yes	characteristic
54	C₂×C₈×C₁₃	M₁₆	1	1	yes	characteristic
55	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
58	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
59	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
60	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
61	C₁₃×C₄⋊C₄	—	2	2	no	—
62	C₁₃×C₂²⋊C₄	—	2	2	no	—
63	C₄²×C₁₃	—	2	1	no	—
64	C₄²×C₁₃	—	2	2	no	—
65	C₄²×C₁₃	C₄²	1	1	yes	characteristic
66	C₄²×C₁₃	C₄²	1	1	yes	characteristic
67	#6090	—	2	2	no	—
68	#6090	—	2	4	no	—
69	#6090	C₄⋊C₄	1	4	yes	—
70	C₂×#364	—	13	1	no	—
71	#2110	—	13	2	no	—
72	C₄×M₃₂	—	13	1	no	—
73	#1772	D₁₃	1	1	yes	characteristic
74	#1378	—	13	2	no	—
75	C₂³×C₁₃	#131	1	1	yes	characteristic
76	C₂×C₄×C₁₃	—	2	1	no	—
77	C₂×C₄×C₁₃	—	2	1	no	—
78	C₂×C₄×C₁₃	—	4	2	no	—
79	C₂×C₄×C₁₃	—	4	2	no	—
80	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
81	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
82	C₂×C₄×C₁₃	C₄×Q₈	1	1	yes	characteristic
83	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
84	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
85	C₂×C₄×C₁₃	#131	1	1	yes	characteristic
86	C₈×C₁₃	—	4	2	no	—
87	C₈×C₁₃	#97	1	1	yes	characteristic
88	C₈×C₁₃	#97	1	1	yes	characteristic
89	C₈×C₁₃	#97	1	1	yes	characteristic
90	C₈×C₁₃	#97	1	1	yes	characteristic
91	C₈×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
92	C₈×C₁₃	#117	1	1	yes	characteristic
93	C₈×C₁₃	C₄×D₄	1	1	yes	characteristic
94	C₈×C₁₃	C₄×Q₈	1	1	yes	characteristic
95	C₂×M₃₂	—	13	1	no	—
96	C₂×M₃₂	—	13	1	no	—
97	C₂×M₃₂	—	13	2	no	—
98	C₂×#108	Dic₁₃	1	1	yes	characteristic
99	#427	Dic₁₃	1	1	yes	characteristic
100	#405	C₂×D₁₃	1	2	yes	—
101	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
102	#364	—	13	2	no	—
103	#364	—	13	2	no	—
104	#347	—	26	4	no	—
105	#346	—	13	2	no	—
106	C₄×C₁₆	—	13	2	no	—
107	#339	Dic₁₃	1	2	yes	—
108	C₂²×C₁₃	—	4	1	no	—
109	C₂²×C₁₃	#405	1	1	yes	characteristic
110	C₂²×C₁₃	#436	1	1	yes	characteristic
111	C₂²×C₁₃	#446	1	1	yes	characteristic
112	C₄×C₁₃	—	2	2	no	—
113	C₄×C₁₃	—	4	1	no	—
114	C₄×C₁₃	—	8	2	no	—
115	C₄×C₁₃	C₄×C₄⋊C₄	1	1	yes	characteristic
116	C₄×C₁₃	C₂×#97	1	1	yes	characteristic
117	C₄×C₁₃	#436	1	1	yes	characteristic
118	C₄×C₁₃	#446	1	1	yes	characteristic
119	C₂×M₁₆	C₂×Dic₁₃	1	2	yes	—
120	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
121	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
122	#117	C₂×Dic₁₃	1	2	yes	—
123	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
124	M₃₂	—	13	2	no	—
125	M₃₂	—	13	2	no	—
126	M₃₂	—	13	4	no	—
127	M₃₂	—	26	2	no	—
128	M₃₂	—	26	1	no	—
129	M₃₂	—	26	1	no	—
130	C₂×C₁₆	—	13	2	no	—
131	C₂×C₁₆	—	13	2	no	—
132	C₂×C₁₆	—	26	2	no	—
133	#108	—	2	1	no	—
134	#108	—	2	1	no	—
135	#107	—	2	1	no	—
136	#106	—	2	1	no	—
137	#97	—	2	2	no	—
138	C₄×C₈	—	2	2	no	—
139	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
140	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
141	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
142	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
143	C₂×C₁₃	—	2	1	no	—
144	C₂×C₁₃	—	2	1	no	—
145	C₂×C₁₃	—	2	1	no	—
146	C₂×C₁₃	#1770	1	1	yes	characteristic
147	C₂²×C₄	—	2	1	no	—
148	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
149	M₁₆	—	2	2	no	—
150	M₁₆	—	2	2	no	—
151	C₂×C₈	—	2	2	no	—
152	C₂×C₈	#6099	1	1	yes	characteristic
153	C₂×C₈	#6099	1	1	yes	characteristic
154	C₂×C₈	#6099	1	1	yes	characteristic
155	C₂×C₈	#6099	1	1	yes	characteristic
156	C₂×C₈	C₄×Dic₁₃	1	1	yes	characteristic
157	C₂×C₈	C₄×Dic₁₃	1	1	yes	characteristic
158	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
159	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
160	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
161	C₂×C₈	#6129	1	1	yes	characteristic
162	C₂×C₈	Q₈×D₁₃	1	1	yes	characteristic
163	C₂×C₈	#6131	1	1	yes	characteristic
164	C₄⋊C₄	—	2	2	no	—
165	C₂²⋊C₄	—	2	2	no	—
166	C₄²	—	2	1	no	—
167	C₄²	—	2	2	no	—
168	C₄²	C₄×Dic₁₃	1	1	yes	characteristic
169	C₄²	C₄×Dic₁₃	1	1	yes	characteristic
170	C₁₆	—	13	4	no	—
171	C₁₆	—	26	2	no	—
172	C₁₆	—	26	4	no	—
173	C₁₃	#12667	1	1	yes	characteristic, complemented, Sylow
174	C₂³	#88202	1	1	yes	characteristic
175	C₂×C₄	—	2	1	no	—
176	C₂×C₄	—	2	1	no	—
177	C₂×C₄	—	4	2	no	—
178	C₂×C₄	—	4	2	no	—
179	C₂×C₄	C₂×#6099	1	1	yes	characteristic
180	C₂×C₄	C₂×#6099	1	1	yes	characteristic
181	C₂×C₄	C₂×C₄×Dic₁₃	1	1	yes	characteristic
182	C₂×C₄	D₄×Dic₁₃	1	1	yes	characteristic
183	C₂×C₄	Q₈×Dic₁₃	1	1	yes	characteristic
184	C₂×C₄	#88202	1	1	yes	characteristic
185	C₈	—	4	2	no	—
186	C₈	#88043	1	1	yes	characteristic
187	C₈	#88043	1	1	yes	characteristic
188	C₈	#88043	1	1	yes	characteristic
189	C₈	#88043	1	1	yes	characteristic
190	C₈	#88141	1	1	yes	characteristic
191	C₈	C₄⋊C₄×D₁₃	1	1	yes	characteristic
192	C₈	#88146	1	1	yes	characteristic
193	C₈	#88147	1	1	yes	characteristic
194	C₂²	—	4	1	no	—
195	C₂²	#11718658	1	1	yes	characteristic
196	C₂²	#11718673	1	1	yes	characteristic
197	C₂²	#11718685	1	1	yes	characteristic
198	C₄	—	2	2	no	—
199	C₄	—	4	1	no	—
200	C₄	—	8	2	no	—
201	C₄	C₂×#88043	1	1	yes	characteristic
202	C₄	C₄⋊C₄×Dic₁₃	1	1	yes	characteristic
203	C₄	#11718673	1	1	yes	characteristic
204	C₄	#11718685	1	1	yes	characteristic
205	C₂	—	2	1	no	—
206	C₂	—	2	1	no	—
207	C₂	—	2	1	no	—
208	C₂	#13114705	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#88146
commutator	C₂×C₄×C₁₃	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₁₃
Fitting	C₁₃×#1772	C₂
socle	C₂×C₁₃	#1770
prime radical	C₂³×C₁₃	#131
cosocle	C₂²×C₈×C₁₃	C₂³
hypercenter	#1772	D₁₃
hypocenter	C₁₃	#12667
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #1772

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
15. #11718055	C₄	semidirect
16. #11718055	C₄	semidirect
17. #11718053	C₄	semidirect
18. C₄×#6090	C₄	semidirect
173. C₁₃	#12667	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#11718055	2
#13116441	2
C₄×#88052	2
#13112922	2
C₁₃×#1772	2
#12667	13